The Death Row Problem

When Should We Expect Mercy?

A version of this article appears in TWISTED LOGIC: Puzzles, Paradoxes, and Big Questions. By Leighton Vaughan Williams. Chapman & Hall/CRC Press. 2024.

THE SETTING

The setting is a prison where three inmates—Amos, Bertie, and Casper—are awaiting the hangman’s noose. The warden, in an act of clemency to celebrate the King’s birthday, will grant clemency to just one of the three prisoners. The choice of which of these inmates to pardon is made randomly, with each name placed in a hat and drawn out. The warden now knows whom he will pardon, but the men on death row do not.

THE REQUEST

Amos makes a request to the warden. He asks the warden to name a prisoner who will NOT be pardoned, without revealing his (Amos’s) own fate. If the warden has chosen Bertie to be granted clemency, he should name Casper as one of the doomed. If it’s Casper who has been pardoned, the warden should name Bertie to be executed. If Amos himself is to be pardoned, the warden should simply toss a coin and name either Bertie or Casper as one of the doomed.

Amos’s Request: It’s essential to note here that Amos’s request is based on the assumption that the warden will not reveal if Amos himself is the pardoned prisoner.

THE WARDEN’S RESPONSE

The warden agrees to the request from Amos and reveals that Casper is not the pardoned prisoner.

WHAT DOES THIS MEAN FOR AMOS AND BERTIE?

With this new information to hand, each of the prisoners can re-evaluate their chances. Initially, Amos believes that his chance of a pardon is 1/3, but with Casper out of the running, he believes that his odds of clemency have risen to 1/2. But is he right in this belief?

RE-EVALUATING THE ODDS

Initially, the odds are 1/3 for each prisoner because only one of the three prisoners is chosen at random to be pardoned. However, when the warden reveals that Casper will not be pardoned, Amos gains new information but not about his own fate. There’s no new information regarding his own fate, so his chances remain as they were, at 1/3. Meanwhile, Bertie’s odds of being pardoned have now increased to 2/3.

WHY ARE THE ODDS DIFFERENT?

This difference in the odds between Amos and Bertie might seem to be counterintuitive. How can they both receive the same information, yet have different survival odds? The answer lies in the warden’s selection process. The warden would not have revealed Amos as the condemned prisoner due to Amos’s unique request, but he might have revealed Bertie as such, instead of Casper. The fact that he doesn’t name Bertie when he might have done so indicates that Bertie’s chances of being pardoned have increased, while nothing has changed for Amos. Amos’s belief that his odds have increased to 1/2 is a misconception.

A LARGER SCENARIO

If this still seems puzzling, consider a larger group of 26 prisoners. If Amos asks the warden to name 24 condemned prisoners in random order without revealing his own fate on any occasion, each prisoner initially has a 1/26 chance of being pardoned. But every time a doomed prisoner is named, the chance that each of the remaining prisoners (except for Amos) will be pardoned increases.

Once every prisoner but Bertie has been named as condemned, Amos’s chances of survival remain at 1/26. However, Bertie’s odds of being pardoned have now increased to 25/26, even though only two prisoners remain unnamed by the warden.

So, even though it might seem like Amos has a very good chance of being pardoned, the reality is that his odds have not changed and remain at 25/1, representing a chance that Amos will escape the noose of 1/26.

CONCLUSION: THE KEY TAKEAWAY

The Three Prisoners Problem highlights the importance of understanding the method by which we obtain information and its impact on the probabilities. It’s a fascinating exploration of conditional probability that shows how the same piece of information can affect the chances of two individuals differently, based on the process by which that information was revealed. As such, it is a classic example of how counterintuitive probability can be, especially in situations where information is revealed in a conditional manner.